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Optimal limits on primordial magnetic fields from CMB temperature bispectrum of passive modes

We investigate bounds on the strength of the primordial magnetic field (PMF) from the cosmic microwave background (CMB) bispectra of the intensity (temperature) modes induced from the auto- and cross-correlated bispectra of the scalar and tensor components of the PMF anisotropic stress. At first, we construct a general formula for the CMB intensity and polarization bispectra from PMFs composed of any type of perturbation. Then we derive an approximate expression which traces the exact shape of the CMB bispectrum in order to reduce the computation time with respect to a large number of the multipole configurations, and also show that the non-Gaussian structure coming from PMFs is classified as the local-type configuration. Computing the signal-to-noise ratio on the basis of the approximate formula with the information of the instrumental noises and resolutions, we find expected upper bounds on the magnetic field strength, when the magnetic spectrum is nearly scale invariant ($n_B = -2.9$), smoothed on $1 {\rm Mpc}$ scale at 95% confidence level from the WMAP and PLANCK experiments as $B_{1 \rm Mpc} < 4.0 - 6.7 {\rm nG}$ and $3.8 - 6.5 {\rm nG}$, respectively, depending on the energy scale of the magnetic field production from $10^{14} {\rm GeV}$ to $10^3 {\rm GeV}$. Our new consequences imply slight overestimations by the previous rough discussions.